Use custom values as units?
Blender only has 1 unitless unit: the Blender Unit.
You can't create any others, but that doesn't matter, since it does exactly what you want.

At first your BU equals 5mm. Enter 0.005 in the Unit Scale checkbox.



Switching between metric and None will you either Dimensions of (1, 5, 3) or (5mm, 25mm, 15mm).

Changing the translating to 1 BU = 6mm seems pretty straightforward. Simply entering the new value 0.006 produces dimensions of (6mm, 30mm 18mm).